Is Space-Time Discrete? Discrete matter At a seaport in the Aegean around the
year 500BC two philosophers, Leucippus and his student Democritus, pondered the
idea that matter was made of indivisible units separated by void. Was it a
remarkable piece of insight or just a lucky guess? At the time there was
certainly no compelling evidence for such a hypothesis. Their belief in the atom
was a response to questions posed earlier by Parmenides and Zeno. Perhaps they
were also inspired by the coarseness of natural materials like sand and stone.
Democritus extended the concept as far as it could go claiming that not just
matter, but everything else from colour to the human soul must also consist of
atoms.

The idea was subsequently supplanted by the very different philosophies of
Plato and Aristotle who believed that matter was infinitely divisible and that
nature was constructed from perfect symmetry and geometry. Matter was composed
of four elements, Earth, Air, Fire and Water.

In the 1660's Robert Boyle, a careful chemist and philosopher was one of the
first to seek a revision. He proposed a corpuscular theory of matter to explain
behaviour of gases such as diffusion. According to Boyle there was only one
element, all corpuscles would be identical. Different substances would be
constructed by combining the corpuscules in different ways. The theory was based
as much on the alchemist's belief in the existence of a philosopher's stone
which could turn lead into gold, as is was on empirical evidence. Newton built
on the corpuscular theory with the mechanistic philosophy of Descartes. He saw
the corpuscles as units of mass and introduced the laws of mechanics to explain
their motion.

In 1811 the atomic theory was again resurrected by John Dalton to explain
chemical composition. Avogadro developed the molecular theory and his law that
all gases at the same temperature, pressure and volume contain the same number
of molecules even though their weights are different. By the mid nineteenth
century the number of molecules could be measured. Maxwell and Boltzmann went on
to explain the laws of thermodynamics through the statistical physics of
molecular motion. Despite this indirect evidence many scientists were sceptical
of the kinetic theory until Einstein supported it. In the early eighteenth
century, A biologist Robert Brown had observed random motion of particles
suspended in gases. Einstein explained that this Brownian motion could be seen
as direct experimental evidence of molecules which were jostling the particles
with their own movements.

How far has modern physics gone towards the ideal of Democritus that
everything should be discrete?

The story of light parallels that of matter. Newton extended Boyle's
corpuscular theory to light but without empirical foundation. Everything he had
observed and much more was later explained by Maxwell's theory of
Electromagnetism in terms of waves in continuous fields. It was Planck's Law and
the photoelectric effect which later upset the continuous theory. These
phenomena could best be explained in terms of light quanta. Today we can detect
the impact of individual photons on CCD cameras even after they have travelled
across most of the observable universe from the earliest moments of galaxy
formation.

Those who resisted the particle concepts had, nevertheless, some good sense.
Light and matter, it turns out, are both particle and wave at the same time. The
paradox is at least partially resolved within the framework of Quantum Field
Theories where the duality arises from different choices of basis in the Hilbert
space of the wave function.

Discrete Space-Time After matter and light, history is repeating itself for a
third time and now it is space-time which is threatened to be reduced to
discrete events. The idea that space-time could be discrete has been a recurring
one in the scientific literature of the twentieth century. A survey of just a
few examples reveals that discrete space-time can actually mean many things and
is motivated by a variety of philosophical or theoretical influences.

It has been apparent since early times that there is something different
between the mathematical properties of the real numbers and the quantities of
measurement in physics at small scales. Riemann himself remarked on this
disparity even as he constructed the formalism which would be used to describe
the space-time continuum for the next century of physics. When you measure a
distance or time interval you can not declare the result to be rational or
irrational no matter how accurate you manage to be. Furthermore it appears that
there is a limit to the amount of detail contained in a volume of space. If we
look under a powerful microscope at a grain of dust we do not expect to see
minuscule universes supporting the complexity of life seen at larger scales.
Structure becomes simpler at smaller distances. Surely there must be some
minimum length at which the simplest elements of natural structure are found and
surely this must mean space-time is discrete.

This style of argument tends to be convincing only to those who already
believe the hypothesis. It will not make many conversions. After all, the modern
formalism of axiomatic mathematics leaves no room for Zeno's paradox of Archiles
and the tortoise. However, such observations and the discovery of quantum theory
with its discrete energy levels and the Heisenberg uncertainty principle led
physicists to speculate that space-time itself may be discrete as early as the
1930's. In 1936 Einstein expressed the general feeling that ... perhaps the
success of the Heisenberg method points to a purely algebraic method of
description of nature, that is, to the elimination of continuous functions from
physics. Then, however, we must give up, by principle, the space-time continuum
.... Heisenberg himself noted that physics must have a fundamental length scale
which together with Planck's constant and the speed of light permit the
derivation of particle masses. Others also argued that it would represent a
limit on the measurement of space-time distance. At the time it was thought that
this length scale would be around 10-15m corresponding to the masses of the
heaviest particles known at the time but searches for non-local effects in high
energy particle collisions have given negative results for scales down to about
10-19m and today the consensus is that it must correspond to the much smaller
Planck length at 10-35m.

The belief in some new space-time structure at small length scales was
reinforced after the discovery of ultraviolet divergences in Quantum Field
Theory. Even though it was possible to perform accurate calculations by a
process of renormalisation many physicists felt that the method was incomplete
and would break down at smaller length scales unless a natural cut-off was
introduced.

A technique which introduces such a minimum length into physics by quantising
space-time was attempted by Snyder in 1947. Snyder introduced non-commutative
operators for space-time co-ordinates. These operators have a discrete spectrum
and so lead to a discrete interpretation of space-time. The model was Lorentz
invariant but failed to preserve translation invariance. Similar methods have
been tried by others since and although no complete theory has come of these
ideas there has been a recent upsurge of renewed interest in quantised
space-time, now re-examined in the light of quantum groups.

Another way to provide a small distance cut-off in field theory is to
formulate it on a discrete lattice. This approach was introduced in 1940 by
Wentzel but only later studied in any depth. If the continuum limit is not to be
restored by taking the limit where the lattice spacing goes to zero then the
issue of the loss of Lorentz invariance must be addressed.

None of these ideas were really very inventive in the way they saw
space-time. Only a rare few such as Finkelstein with his space-time code or
Penrose with twistor theory and spin networks could come up with any concrete
suggestions for a more radical pregeometry before the 1980's.

Another aspect of the quantum theory which caused disquiet was its inherent
indeterminacy and the essential role of the observer in measurements. The
Copenhagen interpretation seemed inadequate and alternative hidden variable
theories were sought. It was felt that quantum mechanics would be a statistical
consequence of a more profound discrete deterministic theory in the same sense
that thermodynamics is a consequence of the kinetic gas theory.

Over the years many of the problems which surrounded the development of the
quantum theory have diminished. Renormalisation itself has become acceptable and
is proven to be a consistent procedure in perturbation theory of gauge field
physics. The perturbation series itself may not be convergent but gauge theories
can be regularised non-perturbatively on a discrete lattice and there is good
reason to believe that consistent Quantum Field Theory can be defined on
continuous space-time at least for non-abelian gauge theories which are
asymptotically free. In Lattice QCD the lattice spacing can be taken to zero
while the coupling constant is rescaled according to the renormalisation group.
In the continuum limit there are an infinite number of degrees of freedom in any
volume no matter how small.

Quantum indeterminacy has also become an acceptable aspect of physics. In
1964 John Bell showed that most ideas for hidden variable theories would violate
an important inequality of quantum mechanics. This inequality was directly
verified in a careful experiment by Alain Aspect in 1982. There are still those
who try to get round this with new forms of quantum mechanics such as that of
David Bohm, but now they are a minority pushed to the fringe of established
physics. Everett's thesis which leads us to interpret quantum mechanics as a
realisation of many worlds has been seen as a resolution of the measurement
problem for much of the physics community.

Without the physical motivation discrete space-time is disfavoured by many.
Hawking says Although there have been suggestions that space-time may have a
discrete structure I see no reason to abandon the continuum theories that have
been so successful. Hawking makes a valid point but it may be possible to
satisfy everyone by invoking a discrete structure of space-time without
abandoning the continuum theories if the discrete-continuum duality can be
resolved as it was for light and matter.

Discreteness in Quantum Gravity It is only when we try to include gravity in
Quantum Field Theory that we find solid reason to believe in discrete
space-time. With quantisation of gravity all the old renormalisation issues
return and many new problems arise. Whichever approach to quantum gravity is
taken the conclusion seems to be that the Planck length is a minimum size beyond
which the Heisenberg Uncertainty Principle prevents measurement if applied to
the metric field of Einstein Gravity.

Does this mean that space-time is discrete at such scales with only a finite
number of degrees of freedom per unit volume? Recent theoretical results from
String Theories and the Loop-representation of Gravity do suggest that
space-time has some discrete aspects at the Planck scale.

The far reaching work of Bekenstein and Hawking on black hole thermodynamics
has led to some of the most compelling evidence for discreteness at the Planck
scale. The black hole information loss paradox which arises from semi-classical
treatments of quantum gravity is the nearest thing physicists have to an
experimental result in quantum gravity. Its resolution is likely to say
something useful about a more complete quantum gravity theory. There are several
proposed ways in which the paradox may be resolved most of which imply some
problematical breakdown of quantum mechanics while others lead to seemingly
bizarre conclusions.

One approach is to suppose that no more information goes in than can be
displayed on the event horizon and that it comes back out as the black hole
evaporates by Hawking radiation. Bekenstein has shown that if this is the case
then very strict and counter-intuitive limits must be placed on the maximum
amount of information held in a region of space. It has been argued by 't Hooft
that this finiteness of entropy and information in a black-hole is also evidence
for the discreteness of space-time. In fact the number of degrees of freedom
must be given by the area in Planck units of a surface surrounding the region of
space. This has led to some speculative ideas about how quantum gravity theories
might work through a holographic mechanism, i.e. it is suggested that physics
must be formulated with degrees of freedom distributed on a two dimensional
surface with the third spatial dimension being dynamically generated.

At this point it may be appropriate to discuss the prospects for experimental
results in quantum gravity and small scale space-time structure. Over the past
twenty years or more, experimental high energy physics has mostly served to
verify the correctness of the Standard Model as proposed theoretically between
1967 and 1973. We now have theories extending to energies way beyond current
accelerator technology but it should not be forgotten that limits set by
experiment have helped to narrow down the possibilities and will presumably
continue to do so.

It may seem that there is very little hope of any experimental input into
quantum gravity research because the Planck energy is so far beyond reach.
However, a theory of quantum gravity would almost certainly have low energy
consequences which may be in reach of future experiments. The discovery of
supersymmetry, for example, would have significant consequences for theoretical
research on space-time structure.

It from Bit In the late 1970's the increasing power of computers seems to
have been the inspiration behind some new discrete thinking in physics. Monte
Carlo simulations of lattice field theories were found to give useful numerical
results with surprisingly few degrees of freedom where analytic methods have
made only limited progress.

Cellular automata became popular at the same time with Conway's invention of
the Game of Life. Despite its simple rules defined on a discrete lattice of
cells the game has some features in common with the laws of physics. There is a
maximum speed for causal propagation which plays a role similar to the speed of
light in special relativity. Even more intriguing is the accidental appearance
of various species of glider which move through the lattice at fixed speeds.
These could be compared with elementary particles.

For those seeking to reduce physics to simple deterministic laws this was an
inspiration to look for cellular automata as toy models of particle physics
despite the obvious flaw that they broke space-time symmetries. The quest is not
completely hopeless. With some reflection it is realised that a simulation of an
Ising model with a metropolis algorithm is a cellular automaton if the
definition is relaxed to allow probabilistic transitions. The Ising model has a
continuum limit in which rotational symmetry is restored. It is important to our
understanding of integrable quantum field theories in two dimensions. 't Hooft
has also looked to cellular automata as a model of discrete space-time physics.
His motivation is somewhat different since indeterminacy in quantum mechanics
is, for him, quite acceptable. He suggests that the states of a cellular
automata could be seen as the basis of a Hilbert space on which quantum
mechanics is formed.

The influence of computers in physics runs to deeper theories than cellular
automata. There is a school of thought which believes that the laws of physics
will ultimately be understood as being a result from information theory. The
basic unit of information is the binary digit or bit and the number of bits of
information in a physical system is related to its entropy.

J.A Wheeler has sought to extend this idea, .. every physical quantity, every
it, derives its ultimate significance from bits, a conclusion which we epitomise
in the phrase, It from Bit . For Wheeler and his followers the continuum is a
myth, but he goes further than just making space-time discrete. Space-time
itself, he argues, must be understood in terms of a more fundamental pregeometry
. In the pregeometry there would be no direct concepts of dimension or
causality. Such things would only appear as emergent properties in the
space-time idealisation.

So is it or isn't it? There do seem to be good reasons to suppose that
space-time is discrete in some sense at the Planck Scale. Theories of quantum
gravity suggest that there is a minimum length beyond which measurement can not
go, and also a finite number of significant degrees of freedom. In canonical
quantisation of gravity, volume and area operators are found to have discrete
spectra, while topological quantum field theories in 2+1 dimensions have exact
lattice formulations.

At the same time, the mathematics of continuous manifolds seem to be
increasingly important. Topological structures such as instantons and magnetic
monopoles appear to play their part in field theory and string theory. Can such
things be formulated on a discrete space?

The riddle will most probably be resolved through a dual theory of space-time
which has both discrete and continuous aspects.